Time-Space Network Representation

5. 4 Time-Space Network Representation

A time-space network representation is often employed in representing rail operations over a period of time. Such a time-space network representation is used in this study to create the rail timetable. Let G = ( N , A ) be a directed network consisting of a set of nodes, N, and a

set of arcs, A. The nodes represent terminals, ports, and border crossings (referred to herein generically as terminals) and arcs represent rail tracks. The arcs connect the terminals. In the node set N, let α

be a pseudo super-source terminal, β be a pseudo super-sink terminal, V denote a set of terminals at which trains may arrive and U denote a set of terminals from

which trains may depart. Node v ∈ V is referred to as an arrival terminal and node u ∈ U is referred to as a departure terminal. In the arc set A, each arc a ∈ A has two endpoints u and v. Thus, whenever a rail track ( u , v ) ∈ A , it is feasible for a train to depart at terminal v and

arrive at terminal u running along the rail track (v u , ) .

We create the time-space network, G= p ( N p , A p ) , over a period of time, T={0,1,…,W}, from network G. The nodes (except for the pseudo super-source and sink

nodes) and arcs in the time-space network p G have both a space and time component. Let the node set t N=

{ U p , V p , α , β } , where node v∈ V p denotes node v of G at time t ∈ T and

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p is an arc representing a potential space-time pair for which a train can depart from terminal u at time r and arrive at

u∈ r U p denotes node u of G at time t ∈ T

. Therefore, q ( u , v ) ∈ A

terminal v by time q.

Let K denote a set of the routes over the time-space network p G . For each route k k ∈ K , let I indicate a set of trains scheduled to be operated along the route within the G.

i = { f , ( f + 1 )..., ( g − 1 ), g } denote an ordered set of terminals that the train i will visit along the route, where

For each train k i ∈ I , let E

f is defined as the route’s origin and g is defined as the route’s destination. Thus, the schedule for each train k i ∈ I on the route k ∈ K (i.e. the

sequence that the train visits the terminals and the train departure and arrival times at each terminal) will be contained in an arc set defined as i A

Based on the notation defined above, the time-space network representation, in which the train departs from its origin, picks up or drops off shipments at some intermediate terminals, and arrives at its destination along the network, is illustrated in Figure 5-2. Within

p G , the train’s itinerary along the route and its activities required at the terminal can be represented by four different types of arcs: i) departure arcs, ii) movement arcs, iii)

process/siding arcs, and iv) arrival arcs. Each arc is described below.

1. De parture arc to represent the train departure from the pseudo super-source node to origin terminal of the train. The departure arc q ( α , u ) corresponds to a feasible departure of the train at the origin.

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2. Movement arc to represent the train traveling on a track between any two intermediate terminals.

A movement arc r ( u , v ) represents the train i traveling from terminal u at time q to another terminal v at time r. Thus, the travel time from u to v will be equal to r minus q.

3. Process/siding arc to represent the train process or siding at the terminal.

A processing/siding arc q ( v , u ) represents either a railcar processing activity at the terminal (i.e. railcar-to-block and block-to-train assignment as well as the locomotive

replacement) before a railcar is carried by train or a train is temporarily held at the terminal. In addition, the border crossing is also modeled at a terminal. The time period associated with the process/siding arc is equal to at least the minimum time period of the railcar processing activity, border crossing or train holding at the terminal.

Sidings en route are not explicitly modeled. Trains are simply not permitted to operate on the same track in the same time slot.

4. Arrival arc to represent train arrival at the destination terminal of the train. An arrival arc r ( v , β ) denotes an arc connecting the destination terminal to the

pseudo super-sink node. Thus, it corresponds to a feasible train arrival at the destination terminal.

To model the operational cost in the objective function of the train slot generation model, a charge is associated with each arc in the time-space network if required. The operational cost of the train consists of the locomotive charge, the track access charge and the infrastructure charge. The locomotive charge is applied to the departure arc q ( α , u ) , the track

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access charge is applied to the movement arc q ( u , v ) , and the infrastructure charge is

applied to the process/siding arc q ( v , u ) . Note that there is no charge associated with the arrival arc r ( v , β ) .

To model shipment delay (actual arrival time minus preferred arrival time) in the objective function, we need to know the arrival time at the destination terminal for each train

i k I v ∈ r . The arrival time can be obtained from the arrival arc ( , β ) . The train i’s arrival time is equal to r, since r is the time when the train departs from v for the pseudo super-sink node

β . If the train arrives later the train’s ideal arrival time (i.e. 5:00 a.m. on Friday morning), delay is incurred. Let µ denote the delay of train i and set it to the difference between the i

train’s actual arrival time and the ideal arrival time. A penalty, ρ ( µ i ) , is imposed on the objective function whenever a train delay µ exists. i

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Time Terminal 1

Departure

Arrival Terminal 2

Departure

Arrival Terminal 3

Departure

Arrival Terminal 4 Departure

Terminal 5 Arrival

Departure arc

Pseudo super-source /sink node

Movement arc

Arrival node

Processing/siding arc

Departure node

Arrival arc

Figure 5-2. A time-space network representation.